Engaging students: Midpoint

In my capstone class for future secondary math teachers, I ask my students to come up with ideas for engaging their students with different topics in the secondary mathematics curriculum. In other words, the point of the assignment was not to devise a full-blown lesson plan on this topic. Instead, I asked my students to think about three different ways of getting their students interested in the topic in the first place.

I plan to share some of the best of these ideas on this blog (after asking my students’ permission, of course).

This student submission again comes from my former student Danielle Pope. Her topic, from Geometry: deriving the term midpoint.

How could you as a teacher create an activity or project that involves your topic?

Introducing the definition of a midpoint in the classroom will take using class time to let students explore for themselves. The activity that I would make my students do is have the entire class stand up and have 2 students stand at opposite sides of the room. I would then ask my students to line up shoulder to shoulder. Once they were in a straight line I would ask “who is perfectly in the middle of this line?” This is where I would give my students 10 minutes initially to come up with various ways of how they would prove a student was in the middle of the line. Various “proofs” that they could tell me would be that there is exactly the same number of people on each side of the middle person. If that answer was given I would make an odd number of students stand in line and ask the same question of “Who is in the middle”? They would have to reconsider this answer because they couldn’t cut the student in half but I would hope that they would come to the conclusion that they would have to half the person in order to find the perfect center. Another “proof” that they may give me is measuring the distance from one end to the other and half that distance to find the person in the middle. This can also start that same conversation of how we would find the exact “midpoint” without cutting the person into pieces.

How has this topic appeared in pop culture (movies, TV, current music, video games, etc.)?

To get just a basic definition of the midpoint, we can look at the lingo used in all sporting events. All sports have some form of a season that lasts for a certain amount of time. For this example specifically, I will be looking at the football season. Towards the middle of the season teams will know what to expect by the end. Most of the stats and predictions for teams are made already by the middle or midpoint of a season. In this article about football it relates to what changes various teams needed to make by the middle of their season. Just in the article itself, it says that “we’re now at the midpoint of the NFL season, and while some things are beginning to take shape, there’s still plenty of football left to be played.” In this context, students can understand that midpoint is being used to describe the middle of a football season. With this knowledge, they can use those context clues and just add the numbers given to them.

One of the most important people in mathematics to date would have to be Euclid. Euclid’s book, The Elements, is still the backbone of all mathematics taught from kindergarten to college. One artist took this book or manual to mathematics and put it in the form of artwork. Crockett Johnson is an artist who bases his work off of mathematics. He takes the complicated proofs, lemmas, and theorem that have been proved and puts those in a form that we see as beautiful. One piece that uses mostly all midpoints titled “Bouquet of Triangle Theorems”. This piece is based off of the many of Euclid’s propositions about triangle just used together in one piece of art. For example “the midpoints of the sides of the large triangle in the painting are joined to form a smaller one.” Giving students a copy of this picture they can find various characteristics given a ruler and other tools that can help them possibly come to this conclusion that Euclid already proved. Crockett’s pieces can also be seen at the Smithsonian so that could show kids that math really does show up everywhere in our world even in unexpected places.